Localization of A Category - Abelian Varieties Up To Isogeny

Abelian Varieties Up To Isogeny

An isogeny from an abelian variety A to another one B is a surjective morphism with finite kernel. Some theorems on abelian varieties require the idea of abelian variety up to isogeny for their convenient statement. For example, given an abelian subvariety A₁ of A, there is another subvariety A₂ of A such that

A₁ × A₂

is isogenous to A (Poincaré's theorem: see for example Abelian Varieties by David Mumford). To call this a direct sum decomposition, we should work in the category of abelian varieties up to isogeny.